We begin our introduction to vector spaces with the concrete example of .

Recall from above that denotes the set of all matrices with real entries, and that the elements of this set are called row resp. column vectors when resp. . Our convention will be to denote as simply . In other words, a real m-dimensional vector will always refer to an real column vector (for reasons of spatial economy, though, when writing an element of in coordinate form, we will often express it as the transpose of a row vector).

From the definition of matrix addition and scalar multiplication, we see that in we can i) add two vectors together, and ii) multiply a vector by a scalar, with the result being a (possibly different) vector in the same space that we started with. In other words,

C1
(Closure under vector addition) Given , .
C2
(Closure under scalar multiplication) Given and , .

Moreover, the space equipped with these two operations satisfies certain fundamental properties. In what follows, , , denote arbitrary vectors in , while represent arbitrary scalars in .

A1
(Commutativity of addition) .
A2
(Associativity of addition) .
A3
(Existence of a zero vector) There is a vector with .
A4
(Existence of additive inverses) For each , there is a vector with .
A5
(Distributivity of scalar multiplication over vector addition) .
A6
(Distributivity of scalar addition over scalar multiplication) .
A7
(Associativity of scalar multiplication) .
A8
(Scalar multiplication with 1 is the identity) .

We should briefly mention why satisfies these properties. First, the definition of matrix addition and scalar multiplication imply [C1] and [C2]. The properties [A1] - [A8], excepting [A3] and [A4],are a consequence of Theorem thm:matalg. The so-called existential properties (referring to the fact they claim the existence of certain vectors) follow by direct observation.

These properties isolate the fundamental algebraic structure of , and lead to the following definition of a vector space over (one of the most central in all of linear algebra).